Probabilistic Diophantine Approximation: Randomness in Lattice Point Counting

ISBN-13: 9783319354651 / Angielski / Miękka / 2016 / 487 str.

Jozsef Beck

Opis

This book gives a comprehensive treatment of random phenomena and distribution results in diophantine approximation, with a particular emphasis on quadratic irrationals.

Recenzje

"The author has written a monograph with new results in a new direction of research in probability and number theory, including algebraic number theory. ... The book is presented in a clear style and will be valuable for researchers in the field." (Oto Strauch, Mathematical Reviews, October, 2015)

"Novel, important, intuitively appealing mathematics and a highly readable account. Summing Up: Highly recommended. Upper-division undergraduates and above." (D. V. Feldman, Choice, Vol. 52 (11), July, 2015)

"Beck's book brings together probability theory, algebraic number theory, and combinatorics to forge new ground. ... the book is accessible to a wide range of mathematicians. ... Beck did a nice job of motivating the questions and explaining the answers." (Darren Glass, MAA Reviews, July, 2015)

Spis treści

Preface.- 1 What is "probabilistic" diophantine approximation?.- 2 Expectation, and its connection with quadratic fields.- 3 Variance, and its connection with quadratic fields.- 4 Proving randomness.- 5 Pell equation, super irregularity and randomness.- 6 More on randomness.- References.- Index.

O autorze

József Beck is Harold H. Martin Professor of Mathematics at Rutgers University.

Blurb

This book gives a comprehensive treatment of random phenomena and distribution results in diophantine approximation, with a particular emphasis on quadratic irrationals. It covers classical material on the subject as well as many new results developed by the author over the past decade. A range of ideas from other areas of mathematics are brought to bear with surprising connections to topics such as formulae for class numbers, special values of L-functions, and Dedekind sums. Care is taken to elaborate difficult proofs by motivating major steps and accompanying them with background explanations, enabling the reader to learn the theory and relevant techniques.

Written by one of the acknowledged experts in the field, Probabilistic Diophantine Approximation is presented in a clear and informal style with sufficient detail to appeal to both advanced students and researchers in number theory.